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Analysis of Control Problems for Stationary Magnetohydrodynamics Equations under the Mixed Boundary Conditions for a Magnetic Field

Author

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  • Gennadii Alekseev

    (Institute of Applied Mathematics FEB RAS, 7, Radio St., 690041 Vladivostok, Russia
    Department of Mathematical and Computer Modelling, Far Eastern Federal University, 690922 Vladivostok, Russia)

Abstract

The optimal control problems for stationary magnetohydrodynamic equations under the inhomogeneous mixed boundary conditions for a magnetic field and the Dirichlet condition for velocity are considered. The role of controls in the control problems under study is played by normal and tangential components of the magnetic field given on different parts of the boundary and by the exterior current density. Quadratic tracking-type functionals for velocity, magnetic field or pressure are taken as cost functionals. The global solvability of the control problems under consideration is proved, an optimality system is derived and, based on its analysis, a mathematical apparatus for studying the local uniqueness and stability of the optimal solutions is developed. On the basis of the developed apparatus, the local uniqueness of solutions of control problems for specific cost functionals is proved, and stability estimates of optimal solutions are established.

Suggested Citation

  • Gennadii Alekseev, 2023. "Analysis of Control Problems for Stationary Magnetohydrodynamics Equations under the Mixed Boundary Conditions for a Magnetic Field," Mathematics, MDPI, vol. 11(12), pages 1-29, June.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:12:p:2610-:d:1166026
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    References listed on IDEAS

    as
    1. Evgenii S. Baranovskii, 2021. "Optimal Boundary Control of the Boussinesq Approximation for Polymeric Fluids," Journal of Optimization Theory and Applications, Springer, vol. 189(2), pages 623-645, May.
    2. Exequiel Mallea-Zepeda & Elva Ortega-Torres & Élder J. Villamizar-Roa, 2016. "A Boundary Control Problem for Micropolar Fluids," Journal of Optimization Theory and Applications, Springer, vol. 169(2), pages 349-369, May.
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    Cited by:

    1. Gennadii Alekseev & Yuliya Spivak, 2024. "Stability Estimates of Optimal Solutions for the Steady Magnetohydrodynamics-Boussinesq Equations," Mathematics, MDPI, vol. 12(12), pages 1-42, June.
    2. Gennadii Alekseev & Olga Soboleva, 2024. "Inhomogeneous Boundary Value Problems for the Generalized Boussinesq Model of Mass Transfer," Mathematics, MDPI, vol. 12(3), pages 1-24, January.

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